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Theorem onexlimgt 43232
Description: For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)
Assertion
Ref Expression
onexlimgt (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem onexlimgt
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 omelon 9684 . . . 4 ω ∈ On
2 onun2 6494 . . . 4 ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ∪ ω) ∈ On)
31, 2mpan2 691 . . 3 (𝐴 ∈ On → (𝐴 ∪ ω) ∈ On)
4 onexomgt 43230 . . 3 ((𝐴 ∪ ω) ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
53, 4syl 17 . 2 (𝐴 ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
6 simp2 1136 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝑎 ∈ On)
7 omcl 8573 . . . . 5 ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
81, 6, 7sylancr 587 . . . 4 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (ω ·o 𝑎) ∈ On)
9 noel 4344 . . . . . . . . . 10 ¬ (𝐴 ∪ ω) ∈ ∅
10 oveq2 7439 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (ω ·o 𝑎) = (ω ·o ∅))
11 om0 8554 . . . . . . . . . . . . . . 15 (ω ∈ On → (ω ·o ∅) = ∅)
121, 11ax-mp 5 . . . . . . . . . . . . . 14 (ω ·o ∅) = ∅
1310, 12eqtrdi 2791 . . . . . . . . . . . . 13 (𝑎 = ∅ → (ω ·o 𝑎) = ∅)
1413eleq2d 2825 . . . . . . . . . . . 12 (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ (𝐴 ∪ ω) ∈ ∅))
1514notbid 318 . . . . . . . . . . 11 (𝑎 = ∅ → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
1615adantl 481 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
179, 16mpbiri 258 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
1817pm2.21d 121 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎)))
1918ex 412 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑎 ∈ On) → (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎))))
2019com23 86 . . . . . 6 ((𝐴 ∈ On ∧ 𝑎 ∈ On) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → (𝑎 = ∅ → Lim (ω ·o 𝑎))))
21203impia 1116 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ → Lim (ω ·o 𝑎)))
22 limom 7903 . . . . . . . . 9 Lim ω
231, 22pm3.2i 470 . . . . . . . 8 (ω ∈ On ∧ Lim ω)
246, 23jctir 520 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)))
25 omlimcl2 43231 . . . . . . 7 (((𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
2624, 25sylan 580 . . . . . 6 (((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
2726ex 412 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (∅ ∈ 𝑎 → Lim (ω ·o 𝑎)))
28 on0eqel 6510 . . . . . 6 (𝑎 ∈ On → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
296, 28syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
3021, 27, 29mpjaod 860 . . . 4 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → Lim (ω ·o 𝑎))
31 simp1 1135 . . . . . 6 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ On)
3231, 8jca 511 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On))
33 simp3 1137 . . . . . 6 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
34 ssun1 4188 . . . . . 6 𝐴 ⊆ (𝐴 ∪ ω)
3533, 34jctil 519 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)))
36 ontr2 6433 . . . . 5 ((𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On) → ((𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎)))
3732, 35, 36sylc 65 . . . 4 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎))
38 limeq 6398 . . . . . 6 (𝑥 = (ω ·o 𝑎) → (Lim 𝑥 ↔ Lim (ω ·o 𝑎)))
39 eleq2 2828 . . . . . 6 (𝑥 = (ω ·o 𝑎) → (𝐴𝑥𝐴 ∈ (ω ·o 𝑎)))
4038, 39anbi12d 632 . . . . 5 (𝑥 = (ω ·o 𝑎) → ((Lim 𝑥𝐴𝑥) ↔ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))))
4140rspcev 3622 . . . 4 (((ω ·o 𝑎) ∈ On ∧ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))) → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
428, 30, 37, 41syl12anc 837 . . 3 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
4342rexlimdv3a 3157 . 2 (𝐴 ∈ On → (∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎) → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥)))
445, 43mpd 15 1 (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  cun 3961  wss 3963  c0 4339  Oncon0 6386  Lim wlim 6387  (class class class)co 7431  ωcom 7887   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by: (None)
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